Need asap in the following problem check that it is...

Name: need asap in the following problem check that it is appropriate to use the normal approximation to the binomial then use the normal distribution to estimate the requested probabilities do you take the
Uploaded: 2024-05-18T18:53:58-08:00
Duration: 9 min 8 s
Channel: Jon Southam
Description: need asap in the following problem check that it is appropriate to use the normal approximation to the binomial then use the normal distribution to estimate the requested probabilities do you take the

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 62% of all customers will take free samples. Furthermore, of those who take the free samples, about 41% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 317 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample? (b) What is the probability that fewer than 200 will take your free sample? (c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.41, while P(sample) = 0.62. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Transcript

00:02 So today we're told that 60 % of all customers will take free samples at a grocery store, probably 0 .62.

00:11 And the probability that they'll buy what they sampled is 0 .41.

00:14 And we're going to suppose that on a given day that we're offering free samples, 317 people walk into the supermarket.

00:22 And so we're going to do some approximations here to the normal approximations of the binomial.

00:28 So we need to first be sure we can do that.

00:31 And to ensure that's the case, we need to have n times p times 1 minus p.

00:40 This value needs to be greater than 10.

00:42 And it is, we take n times p, 317 times 0 .62 times 1 minus 0 .62, we get 70, which is greater than 10.

00:54 So we're good to go.

00:55 We can use the approximation of the binomial.

00:57 That means our mean is going to be n times p.

00:59 That's what, so we take this, 317 times .62, 186 .54, that's the mean, and standard deviation sigma is equal to the square root of n times p times n minus p, and we get 8 .64 of two, there we go, that's our standard deviation for this scenario.

01:27 Because what we're asked to do initially is to find the probability that more than 180 will take your free sample, so the probability that x is greater than 180.

01:36 So because we're dealing with discrete values we want to ensure that we're more than 180 so that what we're going to do is we're going to convert something like 180, something around 180, into a z -score.

01:46 That's what we mean by approximate binomial.

01:48 So the z -transformation is this, x minus mu over sigma.

01:53 So x, we're not going to put in 180 because we're dealing with discrete.

01:59 We want to be sure we're above 180 so we don't count the 180.

02:04 So it's so if you think about this, we want more than that.

02:06 So here's our normal distribution.

02:09 Here's x, the mean of 186.

02:13 Whoops, that's the mean.

02:14 Whoops, that's mu.

02:16 And we want 180, which is roughly here.

02:18 But we want to be sure we don't do anything but more than 180.

02:21 So what we do is we go just a little bit above 180, you know, all this stuff.

02:24 So this x value is actually gonna be 180 .5.

02:29 And this, i mean, over sigma.

02:33 And then when we look that value up, the resulting z -score up in a table we're gonna see this red value but that's not what we want we want everything above that value so we do we're gonna do 1 minus the probability that x is less than or equal to 180 because that's what we're gonna get when we look at that z -score but let me show you what i mean so here we go so there's the z -score 180 .5 into this formula.

03:06 Here the mean is 1 .8654, the standard deviation.

03:11 That's the z -score, negative 1 .856.

03:14 Again, if you look that up in a z -table or in the spreadsheet like i did, norm -best -dist, you put in your z -score there, you get this number, 0 .03.

03:21 But that number is this area up here, we want above that.

03:24 So we do 1 minus this, and the probability of z is less than that z.

03:28 What what that means is the probability, oops, is the probability that z is less than negative 1 .86.

03:36 I'm just rounding there for simplicity's sake.

03:38 But we want more than that.

03:39 The probability that z is greater than negative 1 .86...